1. #post_modern Branch-and-Cut Implementation
Matteo Fischetti, University of Padova
ISMP 2018, Bordeaux, July 6, 2018

2. Why bothering about implementations at ISMP?
ISMP 2018, Bordeaux, July 6, 2018

3. Why bothering about implementations at ISMP?
is not just coding!
Needed if we #orms want to have an impact in practical applications
Ask yourself: would Artificial Intelligence (notably: deep learning) be so successful without gradient-descent algorithms served with their efficient #backpropagation implementations?
ISMP 2018, Bordeaux, July 6, 2018

4. Algorithms without implementation
Proof: omitted as of no interest to the typical MP reader.
Describing an Algorithm without Implementation is like stating a Theorem without Proof
#just_a_computational_conjecture
ISMP 2018, Bordeaux, July 6, 2018

5. Algorithms without implementation
Proof: omitted as of no interest to the typical MP reader.
Describing an Algorithm without Implementation is like stating a Theorem without Proof
#just_a_computational_conjecture
ISMP 2018, Bordeaux, July 6, 2018

6. Branch & Cut ™ A “trademark” of Manfred Padberg and Giovanni Rinaldi
Proposed in the 1990’s for the TSP (and soon extended)
Comes as an algorithm entangled with its implementation
Theorem. Using cuts within an enumerative scheme is good.
Proof. Assume w.l.o.g. a good LP solver. Then apply B&Bound but
make use of families of (problem dependent) globally-valid inequalities
perform efficient exact/heuristic cut separation on the fly
use a data-structure (cut pool) to effectively share cuts among nodes
price variables in a dynamic way (well before branch-and-price!)
alternate row and column generation in a sound way …
suspend a node if “unattractive” …
ISMP 2018, Bordeaux, July 6, 2018

7. Modern B&C implementation
Modern B&C solvers such as Cplex, Gurobi, Express, SCIP etc. can be fully customized by using callback functions
Callback functions are just entry points
in the B&C code where an advanced user
(you!) can add his/her customizations
Most-used callbacks (using Cplex’s jargon)
Lazy constraint: add “lazy constr.s” that should be part of the original model
User cut: add additional contr.s that hopefully help enforcing feasibility/integrality
Heuristic: try to improve the incumbent (primal solution) as soon as possible
Branch: modify the branching strategy …
ISMP 2018, Bordeaux, July 6, 2018

8. Lazy constraint callback
Automatically invoked when a solution is going to update the incumbent (meaning it is integer and feasible w.r.t. current model)
This is the last checkpoint where we can discard a
solution for whatever reason (e.g., because it violates
a constraint that is not part of the current model)
To avoid be bothered by this solution again and again, we can/should return a violated constraint (cut) that is added (globally or locally) to the current model
Cut generation is often simplified by the
fact that the solution to be cut is known
to be integer (e.g., SECs for TSP)
ISMP 2018, Bordeaux, July 6, 2018

9. User cut callback
Automatically invoked at every B&B node when the current solution is not integer (e.g., just before branching)
A violated cut can possibly be returned, to be added (locally or globally) to the current model  often leads to an improved convergence to integer solutions
If no cut is returned, branching occurs as usual
Cut generation can be hard as the point is not integer (heuristic approaches can be used)
User cuts are not mandatory for B&C correctness  being too clever on them can actually slow-down the solver because of the overhead in generating and using them (larger/denser LPs etc.)
ISMP 2018, Bordeaux, July 6, 2018

10. Other callbacks
Branch callback: invoked at the end of each node (even when the LP solution is integer and apparently does not require any cut/branching) and used to impose/customize branching
Incumbent callback: invoked just before updating the incumbent (after the lazy constraint callback) to possibly kill a solution without providing any violated cut
Heuristic callback: used to build new (possibly problem-specific) feasible integer solutions
Informative: to just compute/print internal statistics
etc. etc.
ISMP 2018, Bordeaux, July 6, 2018

11. Application: non-convex MIQP
(based on ongoing work with Michele Monaci, U. Bologna,
and Domenico Salvagnin, U. Padova)
Goal: implement a Mixed-Integer (non-convex) Quadratic solver
Two approaches:
1. start with a continuous QP solver and add enumeration on top of it  implement B&B to handle integer var.s
2. start with a MILP solvers (B&C) and customize it to handle the non-convex quadratic terms  add McCormick & spatial branching
PROS: …
CONS: …
ISMP 2018, Bordeaux, July 6, 2018

12. MIQP as a MILP with bilinear eq.s
The fully-general MIQP of interest reads
and can be restated as
ISMP 2018, Bordeaux, July 6, 2018

13. McCormick inequalities
To simplify notation, rewrite the generic bilevel eq as:
Obviously 
(just replace xy by z in the products on the left)
Note: mc1) and mc2) can be improved in case x=y  gradients cuts
ISMP 2018, Bordeaux, July 6, 2018

14. Spatial branching McCormick inequalities are not perfect
 they are tight only when x and/or y
are at their lower/upper bound
 at some B&C nodes, it may happen that the current (fractional or integer) solution satisfies all MC inequalities but some bilinear eq.s
z = xy are still violated (we call this #bilinear_infeasibility)
 we need a bilinear-specific branching (the usual MILP branching on integrality does not work if all var.s are integer already)
Spatial branching: if z* = x* y* is an offended bilinear eq., branch on
(x ≤ x*) OR (x ≥ x*)
to make the upper (resp. lower) bound on x tight at the left (resp.
right) child node – thus improving the corresponding MC inequality
ISMP 2018, Bordeaux, July 6, 2018

15. Vanilla B&C implementation
Lazy constraint callback: separation of MC inequalities
Usercut callback: not needed (and sometimes detrimental)
Branch callback: spatial branching on the “most offended” z = xy
Incumbent callback: very-last resort to kill a bilinear-infeasible integer solution (when everything else fails e.g. because of tolerances)
Precision: LP precision higher (more restrictive) than bilinear tolerance
MILP heuristics (kindly provided by the MILP solver): active at their default level
MIQP-specific heuristics: not implemented
Implemented but not used in the vanilla version:
additional bilinear-specific cuts  Balas’ Intersection Cuts (ICs)
semi-spatial branching (branch threshold x*+δ  x* violates the x-bound in one of the two children, MC only needed in the other one)
ISMP 2018, Bordeaux, July 6, 2018

16. Does it work?
Comparison with the SCIP 5.0 MIQP solver using CPLEX 12.8 as LP solver + internal nonlinear solver
Preliminary test on the quadratic MINLPlib (700+ instances) …
… but some instances removed as root LP was unbounded
 they need bound tightening by preprocessing (TODO)
Results of our B&C callback-based vanilla implementation using CPLEX 12.8 as MILP solver; 1-thread runs (parallel runs not allowed in SCIP); only instances solved by both codes in the 1-hour time limit.
Overall, we are as fast as SCIP (but the latter solves more instances within the time limit  SCIP qualifies as a more robust solver).
We are 2 to 10 times faster than SCIP when the optimal/best-known solution from MINLPlib is used as a warm-start for both codes  evidently, we miss a sound bilinear-specific heuristic (TODO)
ISMP 2018, Bordeaux, July 6, 2018

17. More detailed comparison
SCIP vs noic (our “vanilla”
version with no ICs and
classical spatial branching) 
Results with incumbent warm-start (only instances solved by both codes)
ISMP 2018, Bordeaux, July 6, 2018

18. Thanks for your attention!
Slides available at .
ISMP 2018, Bordeaux, July 6, 2018